\\ Pari/GP code for working with number field 21.13.6782764925506469650693833692003522155480498176.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^21 - 6*y^19 - 4*y^18 - 126*y^17 - 168*y^16 + 322*y^15 + 756*y^14 + 4554*y^13 + 10912*y^12 + 5454*y^11 - 13020*y^10 - 60139*y^9 - 164556*y^8 - 259701*y^7 - 251642*y^6 - 171396*y^5 - 97416*y^4 - 48624*y^3 - 18144*y^2 - 4032*y - 384, 1)

\\ Defining polynomial: 
K.pol

\\ Degree over Q: 
poldegree(K.pol)

\\ Signature: 
K.sign

\\ Discriminant: 
K.disc

\\ Ramified primes: 
factor(abs(K.disc))[,1]~

\\ Integral basis: 
K.zk

\\ Class group: 
K.clgp

\\ Narrow class group: 
bnfnarrow(K)

\\ Unit rank: 
K.fu

\\ Generator for roots of unity: 
K.tu[2]

\\ Fundamental units: 
K.fu

\\ Regulator: 
K.reg

\\ Analytic class number formula: 
\\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^21 - 6*x^19 - 4*x^18 - 126*x^17 - 168*x^16 + 322*x^15 + 756*x^14 + 4554*x^13 + 10912*x^12 + 5454*x^11 - 13020*x^10 - 60139*x^9 - 164556*x^8 - 259701*x^7 - 251642*x^6 - 171396*x^5 - 97416*x^4 - 48624*x^3 - 18144*x^2 - 4032*x - 384, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

\\ Intermediate fields: 
L = nfsubfields(K); L[2..length(L)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])